Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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45 views

Square integrability of functions

Suppose that for a function $f(x)\,\,, x\in\mathbb{R}$ holds \begin{align} \int_{0}^{T}|f(x)|^{2} ~\mathrm{d}x<\infty \end{align} Does it also holds that \begin{align} ...
2
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1answer
58 views

How to arrive at desired equality?

Why is the following second equality true? $$e^{1+1/2+...+1/(n+1)} - e^{1+1/2+...+1/n} \\= ...
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0answers
42 views

Stuck on continuity proof (like 8 sheets of A4…) $p_if$ is cont. iff $f$ is cont, $p_i:X\rightarrow X_i$ given by $p_i(a)=a_i$ for $a=(a_1,…,a_n)$

Let $Y$ be a metric space, let $f:Y\rightarrow X$ where $(X,d)$ is a metric space given by $X=\prod^n_{i=1}X_i$ equipped with the stadard metric ($\max$) I wish to prove $f$ is continuous iff ...
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1answer
33 views

If each uncountable set $T$ has a countable subset, can we form $T$ by a union of countable subsets?

I was working my way through the set theory chapter in my Analysis textbook when I stumbled across these two theorems: Every infinite set has a countable subset A union of countable subsets ...
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1answer
20 views

Lemniscate curve parametrization exercise

Let $\gamma (t) : \mathbb R \to \mathbb R^2$ be the function $$\gamma(t)=\left(\frac{(1+t^2)t}{1+t^4},\frac{(1-t^2)t}{1+t^4}\right)$$ Prove that the function is $\gamma$ is differentiable, regular ...
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1answer
35 views

Arc lenght of a curve is finite

Let $b<0<a$, and consider the function $\alpha:(0,+\infty) \to \mathbb R^2$ defined as $$\alpha(t)=(ae^{bt}\cos(t),ae^{bt}\sin(t))$$ Show that $\lim_{t \to +\infty} \alpha'(t)=(0,0)$ and ...
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1answer
32 views

Is there an explicit polynomial form for the product of consecutive integers?

I have the product $\prod_{j=0}^{r-1} (n+j)= n(n+1)\cdots(n+r-1)$ where n is a positive integer, and I was wondering if there was an explicit polynomial form for it (as a polynomial of degree r). I've ...
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0answers
57 views

simultaneous trigonometric equations

Consider the pair of simultaneous equations $p_5\cos(2\omega\tau)+p_4\omega\sin(2\omega\tau) = p_1\omega^2-p_3-p_6\cos(3\omega\tau) $, $p_4\omega\cos(2\omega\tau)-p_5\sin(2\omega\tau) = ...
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5answers
88 views

Assumptions in Word Problems (Calculus)

I just had a small question about assumptions in mathematical word problems. Suppose you are given a calculus problem (related-rates), "A spherical balloon is inflated with gas at the rate of 800 ...
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1answer
125 views

How to find this integral $\int_{0}^{\infty}\dfrac{f(x)}{g(x)}dx$ [duplicate]

show that: $$I=\int_{0}^{\infty}\dfrac{x^8-4x^6+9x^4-5x^2+1}{x^{12}-10x^{10}+37x^8-42x^6+26x^4-8x^2+1}dx=\dfrac{\pi}{2}$$ I found this : ...
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13 views

Coordinate Change Operator

Let $ f: \mathbb{R} \rightarrow \mathbb{R} $ be analytic. Recall that for $ h \in \mathbb{R} $, the translated function $ \tilde{f} (x) = f(x+h) $ can be formally written as $ \tilde{f} = e^{ h ...
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1answer
18 views

About a convergence of measurable functions

Let $f_{n}$ be a sequence of measurable functions in M(X,m), is that true that {$ {x∈X∣lim f_{n}∈R}$} $ $ = $⋃ _{M=1} ^∞⋂ _{N=1} ^∞ ⋃ _{n=N}^ ∞ ${x∈X∣ ∣f_{n} -f_{N} ∣< (1/M)}$ $ and that ...
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1answer
16 views

Is this a B-measurable funtion?

Is the function defined by: $f(x)=e^x $ if x is in E and $f(x)=e^{-x} $ if x is not in E measurable?, here f goes from R to R, and E is not member of the Borel-algebra
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2answers
49 views

Some special Metric on R

Apart from usual and discrete metric is there a metric on R which satisfy: d(x, y) = d(x+r , y+r) where x and y are any real no. and r is arbitary real no. Similarly is there a ...
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1answer
40 views

Prove that $f(x) > g(x)$ where both functions are convex and have the same value and slope at $0$

Let $f: [-a,a] \to \mathbb{R}$ and $g: [-a,a] \to \mathbb{R}$ be two non-negative, convex and smooth functions. We further know $f(0) = g(0)=0$ and $f'(0) = g'(0)=0$. I'd like to show $$f(x) \ge ...
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2answers
56 views

$2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$ \hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx. $$ However, in class my teacher defines it without ...
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2answers
53 views

How to check for convexity of function that is not everywhere differentiable?

I have a question. I have just been introduced to the subject of convex sets and convex functions. I read this in wikipedia that a practical test for convexity is - to check whether the 2nd ...
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1answer
71 views

Proving a set of numbers has arithmetic progressions of arbitrary length, but none infinite

For each real number $x$, let $[x]$ be the largest integer less than or equal to $x$. For example, $$[5] = 5$$ $$[7.9] = 7,$$ and $$[−2.4] = −3.$$ An arithmetic progression of length $k$ is a ...
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1answer
71 views

Functions for which $\mathcal{F}g = f \ast f$

Suppose one is given $f \in L^{2}(\mathbb{R})$, my question is whether or not there exists a $g \in L^{1}(\mathbb{R})$ such that $f \ast f = \mathcal{F}g$ where $\mathcal{F}$ is the Fourier transform. ...
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2answers
29 views

How to show $\textrm{supp}(f*g)\subseteq \textrm{supp}(f)+\textrm{supp}(g)$?

Let $f, g\in C_0(\mathbb R^n)$ where $C_0(\mathbb R^n)$ is the set of all continuous functions on $\mathbb R^n$ with compact support. In this case $$(f*g)(x)=\int_{\mathbb R^n} f(x-y)g(y)\ dy,$$ is ...
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1answer
37 views

Prove limit of function

Let $f:(a,+\infty) \to \mathbb{R}$ and on every finite $(a,b)$ interval function $f$ is bounded. Then $$\lim_{x \to \infty}\frac{f(x)}{x}=\lim_{x \to \infty}f(x+1)-f(x)$$ How can we prove or ...
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1answer
96 views

Nonstandard complex numbers and categoricity

Let ${}^*\mathbb{C}$ be a nonstandard complex number field (given, for instance, as a countable ultrapower.) By the transfer principle ${}^*\mathbb{C}$ is algebraically closed of characteristic zero, ...
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2answers
55 views

How should I calculate $\displaystyle\int_{-\infty}^\infty\exp\left\{-\frac{1}{2}(x-it)^2\right\}dx$?

I've read that the residue theorem would help to calculate $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\exp\left\{-\frac{1}{2}(x-it)^2\right\}}_{=:f(x)}dx$$ Since $f$ is an entire function ...
2
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2answers
105 views

A continuously differentiable function with vanishing determinant is non-injective?

(This question relates to my incomplete answer at http://math.stackexchange.com/a/892212/168832.) Is the following true (for all n)? "If $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is continuously ...
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0answers
53 views

Comparing a number with a line of power

How do you compare which is bigger (or maybe equal), LHS or RHS, in $$a \sim b_1^{b_2^{.^{.^{.^{b_n}}}}}$$ given $a$ and $b_i$, $1 \leq i \leq n$, are non-negative integers (also could be big)? The ...
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1answer
33 views

Entire functions such that $\limsup_{z \rightarrow \infty}\frac{|\log |f(z)||}{|z|} < \infty$

The problem I am working on is to find all entire functions satisfying $|f(z)| > 0$ for $|z|$ large and $$\limsup_{z \rightarrow \infty}\frac{|\log |f(z)||}{|z|} < \infty.$$ My guess is that ...
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55 views

Condition on Equality of closure of open ball and closed ball, suppose I have a counterexample

Let $(X, d)$ be a metric space. Also for $x \in X$ and $r \ge 0$ define: $$ B(x,r) = \{ y \in X : d(x,y) < r \} \quad \mbox{ and } \quad K(x,r) = \{ y \in X : d(x,y) \le r \}. $$ Denote by ...
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0answers
17 views

Is $d\mu = g\, dm$ a Radon measure for $g \in L^{1}(dm)$?

Let $m$ is the standard Lebesgue measure and let $g \in L^{1}(dm)$. We know that $m$ is a Radon measure. Is $\mu$ defined by $d\mu = g\, dm$ also a Radon measure? We first claim that $\mu(K) < ...
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1answer
32 views

Dense subset of $L^{2}$ such that $x^{-1/2}f \in L^{1}$ and $\int_{[0, 1]}x^{-1/2}f\, dx = 0$

Does there exist a dense set of functions $f \in L^{2}([0, 1])$ such that $x^{-1/2}f(x) \in L^{1}([0, 1])$ and $\int_{0}^{1}x^{-1/2}f(x)\, dx = 0$? I've noticed that $\int_{0}^{1}x^{-1/2}f(x)\, dx = ...
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1answer
11 views

Bound on the derivative of a cut-off function

Let $\rho$ be a smooth function in $\mathbb R^n$ such that $0 \leq \rho \leq 1$ and $\rho$ is supported in the unit disk and let $\rho_\epsilon(x) = \epsilon^{-n}\rho(\epsilon^{-1}\|x\|)$. If $f$ is ...
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1answer
23 views

Products of distributions ill-defined

This question concerns distributions, as often encountered in PDE theory, which are defined as continuous linear functionals on the space $C_0^{\infty}(\Omega)$ of test functions. The product of two ...
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35 views

Recommendation of analysis textbooks

I'm studying real&complex analysis, functional analysis, measure theory, and Lebesgue integral. I want to make sure which books to use in order to study these fields most rigorously and deeply. ...
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1answer
67 views

Continuity, and continuity in topology.

Metric spaces: Neighborhood of a point $a$ is a Set of point $N$, such that $\exists\delta>0:B_\delta(a)\subset N$ ($B_r(x)$ = open ball at x of radius r) Definition of open set: "A subset $O$ of ...
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5answers
96 views

Prove that if $n$ is an integer, then $n^2 + n^3$ is an even number

I am trying to work through some of the problems in Stephen Lay's Introduction to Analysis with Proof before my Real Analysis class in the fall term starts, and I was just wondering if I could get ...
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1answer
21 views

Proof, that equation decribes trace of curve, which is supposed to be simple

The equation, representing the trace of the curve $$ \varphi(x) = (\cos^3(t), \sin^3(t)) $$ is $1 = x^{\frac{2}{3}} + y^{\frac{2}{3}}$. Proof: Let $(x,y) = (\cos^3 t, \sin^3 t)$, then $x^{1/3} = ...
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1answer
21 views

Ways to represent functions of bounded variation as the difference of two monotone functions

I learnt (it is called Jordan's representation theorem) that every function $f : [a,b] \to \mathbb R$ with bounded variation could be represented as the difference $f = g - h$ of two monotone ...
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1answer
115 views

Understanding last step of a proof that “two trajectories cannot cross at a finite value of t” (Phase trajectories/nodes)

Note: This proof prefaced critical points at the origin for coupled first order ODEs. It was done before showing the asymptotically stable and unstable critical points: Improper, Proper, Spiral, ...
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1answer
27 views

Discretization of an integral

Given $f: [a,b] \to R $ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R $ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ...
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1answer
65 views

Does it converges? $ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $

As I checked on Wolfram Alpha I know that $$ \sum \frac{\sqrt{n}\ln(n)}{n^2+1} $$ Converges. But have tried many tests to show that, without success. I tried ratio/root (inconclusive). Cauchy test ...
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1answer
55 views

Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates

Consider the mapping $\mathbb{R}^3 (x,y,z)$ to $\mathbb{R}^3(u,v,w)$ given in coordinates $$ \left\{ \begin{array}{c} u = yz\sin(x)\\ v = y^2 - x\\ w = xz \end{array} \right. $$ Determine the ranks ...
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3answers
47 views

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping f can not be one-to-one mapping.

$f: \mathbb{R}^2 \to \mathbb{R}$ is a continuously differentiable function (of class $C^1$). Show that mapping $f$ cannot be one-to-one mapping. Let $D_1F(x,y) \neq 0$ for all $(x,y)$ for some open ...
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0answers
17 views

Calderón reproducing formula : $\int_{0}^{\infty}\int_{R^d}|\phi_{t}(x-y)||(\phi_t*f)(y)|\frac{dt}{t}dy<\infty$

Suppose that $\int f=0$, $f \in L^2$ and $f$ has a compact support. Let $\phi$ be radial, and such that $\mathrm{supp}(\phi) \in B(0,1)$. Plus, assume that $\int_{R^+} ...
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30 views

Sum of function applied to parts not equal to function of total

The general goal is to determine the effectiveness of the test pill's ability to keep the test subjects from getting sick using the following data. | Test Subjects | Took Test Pill | ...
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1answer
23 views

Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$

Let $Ω=\{z=x+iy∈C : |y|<x\}.$ Find sup$\{|f′(3)| : f$ maps $Ω$ analytically into the unit disk $\}.$ Okay. So I can find a conformal map from $Ω\rightarrow \mathbb{D}$. I used the map $f(z) = ...
3
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1answer
27 views

Solution of differential equations with discontinuity

Suppose that we have scalar differential equation \begin{equation} \dot{x}(t)=u(t) \end{equation} Here $u(t)$ is a piecewise constant function with discontinuity. If the points of discontinuity is ...
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1answer
32 views

Definition of a metric-nonnegativity condition

There is a question in my mind which seems to be silly but I am desperately wanting the answer. Why a metric is defined from $X\times X$ to $\mathbb R$ and not to the set of nonnegative reals? I ...
0
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1answer
34 views

Prove that $ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$

Let $C_r$ be the circle centered on $0$ with radius $r$ and $t\in \mathbb{R}$. How to show that $$ \frac{1}{2\pi i}\int_{C_r}\frac{e^{\lambda t}}{\lambda^{k+1}}d\lambda =\frac{t^k}{k!}$$
2
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1answer
23 views

Proving a particular subset of a Hilbert space is a subspace

I have a small question please, how to prove that this set: $F=\lbrace h\in H, \langle f''(u)h,h\rangle <0\rbrace$ is a sub space of the Hilbert space $H$, where $f''(u)$ is a self-adjoint ...
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0answers
31 views

Spectral decomposition of a Hilbert space

I have this proof, but I don't understand how they do the spectral decomposition of $H$ into $H_-$and $H_+$? Please help me. Thank you.
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0answers
18 views

Can I calculate the overall % each drug is prescribed if the list of drugs shown to each respondent varies?

I have a survey where the Q's are: Q1: Which of these drugs are you aware of - list of 10 drugs shown Q2: What % of your patients receive each drug (only drugs identified at Q1 shown). As the drugs ...